In mathematics, solid partitions are natural generalizations of partitions and plane partitions defined by Percy Alexander MacMahon. A solid partition of
Contents
- Ferrers diagrams for solid partitions
- Equivalence of the two representations
- Generating function
- Exact enumeration using computers
- Asymptotic behavior
- References
and
Let
Ferrers diagrams for solid partitions
Another representation for solid partitions is in the form of Ferrers diagrams. The Ferrers diagram of a solid partition of
For instance, the Ferrers diagram
where each column is a node, represents a solid partition of
Equivalence of the two representations
Given a Ferrers diagram, one constructs the solid partition (as in the main definition) as follows.
LetGiven a set of
For example, the Ferrers diagram with
with all other
Generating function
Let
The generating functions of partitions and plane partitions have simple formulae due to Euler and MacMahon respectively. However, a guess of MacMahon fails to correctly reproduce the solid partitions of 6 as shown by Atkin et al. It appears that there is no simple formula for the generating function of solid partitions. Somewhat confusingly, Atkin et al. refer to solid partitions as four-dimensional partitions as that is the dimension of the Ferrers diagram.
Exact enumeration using computers
Given the lack of an explicitly known generating function, the enumerations of the numbers of solid partitions for larger integers have been carried out numerically. There are two algorithms that are used to enumerate solid partitions and their higher-dimensional generalizations. The work of Atkin. et al. used an algorithm due to Bratley and McKay. In 1970, Knuth proposed a different algorithm to enumerate topological sequences that he used to evaluate numbers of solid partitions of all integers
which is a 19 digit number illustrating the difficulty in carrying out such exact enumerations.
Asymptotic behavior
It is known that from the work of Bhatia et al. that
The value of this constant was estimated using Monte-Carlo simulations by Mustonen and Rajesh to be